Perfect Numbers

perfect numbers

A perfect number is a special type of number which has relationship with its factors.

These numbers are more important in pure mathematics than in daily life. Their real value lies in helping us understand prime numbers, divisors, mathematical patterns, and number theory.

Perfect Number Definition, Examples

Before we get to the definition of Perfect Numbers, it is important to know what are Proper Factors.

Proper factors are all the factors of a number except the number itself.

A number, the sum of whose factors (except the number itself) is equal to the number is called a perfect number

For example, consider the number 6. The factors of 6 are:

1, 2, 3 and 6

Its proper factors are:

1, 2 and 3

Now add them:

1 + 2 + 3 = 6

Since the sum of the proper factors is equal to the original number, 6 is a perfect number.

It is called a perfect number because its factors balance the number perfectly. The sum is neither less than the number nor greater than the number.

The first few perfect numbers are:

6

28

496

8128

33,550,336

Perfect Numbers Properties

Given below are properties of perfect numbers

  1. All known perfect numbers are even
  2. Every even perfect number is also a triangular number.
  3. Perfect Numbers grow very rapidly – After 8128, the next perfect number is: 33,550,336. This shows that perfect numbers become extremely large very quickly.

Formula for Even Perfect Numbers

Euclid-Euler theorem states that that every even perfect number has the form:

2p-1 ✖️ (2p-1)

where p and 2ᵖ − 1 both must be a prime number.

Prime numbers of the form 2ᵖ − 1 are called Mersenne primes.

If 2ᵖ − 1 is prime, then p must be prime. However, every prime value of p does not necessarily make 2ᵖ − 1 prime.

Example using p = 2

2ᵖ − 1 = 2² − 1 = 3

3 is a prime number.

Now:

2²⁻¹ × 3

= 2¹ × 3

= 2 × 3

= 6

Therefore, 6 is a perfect number.

Example using p = 3

2³ − 1 = 7

7 is a prime number.

Therefore:

2³⁻¹ × 7

= 2² × 7

= 4 × 7

= 28

So, 28 is a perfect number.

Important condition in the formula

It is important to remember that simply taking any prime value of p does not always give a perfect number.

The number:

2ᵖ − 1

must itself also be prime.

For example, take:

p = 11

Then:

2¹¹ − 1 = 2047

But:

2047 = 23 × 89

So, 2047 is not a prime number.

Therefore, p = 11 does not generate a perfect number using this formula.